UAV Swarm-Enabled Aerial CoMP: A Physical Layer Security Perspective

Unlike aerial base station enabled by a single unmanned aerial vehicle (UAV), aerial coordinated multiple points (CoMP) can be enabled by a UAV swarm. In this case, the management of multiple UAVs is important. This paper considers the power allocation strategy for a UAV swarm-enabled aerial network to enhance the physical layer security of the downlink transmission, where an eavesdropper moves following the trajectory of the swarm for better eavesdropping. Unlike existing works, we use only the large-scale channel state information (CSI) and maximize the secrecy throughput in a whole-trajectory-oriented manner. The overall transmission energy constraint on each UAV and the total transmission duration for all the legitimate users are considered. The non-convexity of the formulated problem is solved by using max-min optimization with iteration. Both the transmission power of desired signals and artificial noise (AN) are derived iteratively. Simulation results are presented to validate the effectiveness of our proposed power allocation algorithm and to show the advantage of aerial CoMP by using only the large-scale CSI.


I. INTRODUCTION
In recent years, unmanned aerial vehicles (UAVs) have attracted great interest in wireless communications [1]- [4].Due to their mobility and elevated position, they can provide agile communications [5].With their high maneuverability, UAV can augment the network capacity and coverage, especially in the extreme environments without infrastructure, such as disaster rescue, traffic monitoring and so on [1], [6].More specifically, UAVs are usually cost-effective [7]- [10].They can be exploited to assist on-demand missions, such as highspeed data transmission in the fifth generation (5G) wireless networks.In addition, with the huge demand in emergency applications, i.e., public safety, delivery and surveillance, deploying a flock of UAVs, or swarm, is becoming more attractive, which plays a vital role in meeting performance requirements for communications between multiple UAVs and 5G [11]- [15].
One of the serious concerns in UAV swarm-enabled aerial networks is how to guarantee the privacy and secrecy of the system.Due to the broadcast nature and inherent randomness of wireless channels, UAV swarm-enabled communication networks are particularly vulnerable to various security threats, such as information eavesdropping, information leakage, data modification and so on.In addition, to facilitate the secure transmission, the UAV swarm often places itself near the legitimate users, which is beneficial to eavesdropping, especially when the eavesdropper moves close to the legitimate users.

A. Related Work
To achieve perfect security, the conventional encryption schemes are typically implemented at the upper layer using cryptographic methods.However, this is often achieved at the cost of high computational complexity [16].
Unlike the traditional cryptographic methods, physical layer security (PLS), using the information-theoretic and signal processing approaches, has been widely investigated in the UAV-enabled wireless networks [17]- [20].They enhance the coverage and security of the wireless systems by exploiting physical characteristics of the wireless channel.Specifically, by adaptively adjusting the UAVs' location, they could overcome the propagation constraints in the cellular systems, and provide new possibilities or opportunities for security enhancement.The authors in [17] utilized UAV as a mobile relay, and maximized the secrecy rate of the system with transmit optimization in a four-node.In [18], the authors investigated UAV-enabled secure communication systems where a mobile UAV sent confidential messages to multiple ground users.By considering the imperfect information on the locations of the eavesdroppers, the authors in [19] investigated a UAV-ground communication system with multiple potential eavesdroppers on the ground.The authors in [20] considered UAV-assisted secure communications between a legitimate transmitter-receiver pair for unknown eavesdropper location by taking UAV as an air-to-ground friendly jammer.
These studies [17]- [20] have provided insightful results for improving the secrecy performance of the UAV-aided wireless communications.However, they assume an ideal free-space path-loss model [17]- [19] between the UAV and the legitimate receivers/eavesdroppers or the instantaneous channel state in-formation (CSI) [20] of the eavesdroppers at the transmitter, which may not be practical.
In practice, it is generally difficult to acquire the instantaneous CSI of the eavesdroppers, especially when they are passive.To deal with that, an effective approach, named as artificial noise(AN), has been proposed to mask the desired signals for enhancing the secrecy performance [21]- [25], where AN is designed based on the instantaneous CSI of the legitimate receiver and transmitted in the null-space of the legitimate channel.Although this scheme is helpful for the security, it requires the perfect instantaneous CSI between the source and the legitimate receiver at the transmitter, which is nearly unworkable.The idea is then generalized to the UAV-enabled wireless systems, where a UAV is applied as a mobile jammer to transmit AN [26] or a legitimate receiver [27].However, these works haven't shown useful guidelines to improve physical layer security of UAV swarm-enabled aerial networks.

B. Main Contributions
Despite of the above fruitful results, some challenges still remain in the UAV swarm-enabled aerial networks.
For the UAV swarm-enabled aerial networks, an open challenge is how to acquire CSI.To practically depict the typical propagation environments, the composite channel model, consisting of both small-scale and large-scale fading, needs to be used, which is in stark contrast to the existing literatures [17]- [19].Under the composite channel, one key role for the power allocation strategy is the prior knowledge.Since it is impossible to perfectly acquire the random small-scale fading prior to the whole trajectory of the UAV swarm, it is almost infeasible to assume perfect CSI.In this paper, we devote to guarantee the secrecy performance of the system in a wholetrajectory-oriented manner by utilizing only the large-scale CSI of the legitimate receivers/eavesdroppers, which can be achieved at much lower cost.
In wireless communication systems, path loss could significantly reduce the signal reception quality at the legitimate users, especially in the UAV swarm-enabled aerial networks.In the existing literatures, one effective scheme to overcome the limitation is by means of multiple antenna systems, i.e., multiple-input single-output (MISO) [28], [29], multiple-input multiple-output (MIMO) [30], [31], or single-input multipleoutput (SIMO) [32].However, due to the limited size, it is hard for UAVs to be equipped with multiple antennas.To handle that, we consider an effective coordinated multiple points (CoMP) between UAVs in this paper, where multiple single-antenna UAVs are combined to form the UAV swarm and then act as a virtual multiple-antenna node.Unlike the conventional CoMP with fixed base stations (BSs), the UAV swarm is able to cooperatively operate as an aerial CoMP by utilizing the mobility of the UAVs.Note that, in contrast to the existing works achieving CoMP based on perfect CSI [33], [34], our scheme uses only the predictable large-scale CSI between UAVs and the legitimate receivers/eavesdroppers.
The energy constraint at each UAV is another challenge for the secrecy performance of UAV swarm-enabled aerial networks.Since it's generally difficult to recharge the battery of the UAV during its flight, not only the power budget but also the total energy constraint should be taken into account for each UAV.
Motivated by the above observations, we investigate the AN-aided secure transmission for the UAV swarm-enabled aerial CoMP, where both of the legitimate receivers and eavesdroppers are equipped with multiple antennas.Different from the conventional eavesdropping, we assume the eavesdropper randomly walks following the trajectory of the UAV swarm, which may significantly deteriorate the secrecy performance of the system.In addition, unlike the existing AN-aided secure transmission based on the instantaneous CSI of the legitimate users, AN in our proposed scheme is designed by using only the large-scale CSI.To the best of our knowledge, this is the first time that studies AN-assisted secure transmission in the UAV swarm-enabled aerial CoMP by exploiting only the largescale CSI of the legitimate receivers/eavesdroppers.
Our main contributions of the paper are summarized as follows: • We consider physical layer security in the UAV swarmenabled aerial networks.Specifically, multiple singleantenna UAVs perform an aerial CoMP, and enable a virtual MIMO transmission link with the multiple-antenna legitimate receivers or the eavesdropper, in which the swarm transmits the confidential messages in conjunction with AN and sequentially hovers to serve the scheduled legitimate users.Unlike the existing wiretap mode where the eavesdropper keeps static at a fixed location, we consider the eavesdropper moves following the trajectory of the swarm for better eavesdropping in a passive manner.• To characterize the typical propagation environments, we consider a practical composite channel model consisting of both small-scale and large-scale fading.However, it is infeasible to achieve perfect CSI since the small-scale channel fading is time-varying and hard to be acquired.
In this work, we use only the large-scale channel fading, which is more reasonable because the large-scale channel fading mainly depends on the position information of both the UAV and the legitimate receivers/eavesdroppers.
We can obtain such information based on the historical data and the related distance between the UAV and the legitimate receivers/eavesdroppers. • An optimization framework in a whole-trajectoryoriented manner is proposed to maximize the ergodic secrecy rate (ESR) by jointly optimizing the power allocation between the confidential messages and AN under the overall energy budget at each UAV.The formulated problem is not convex and hard to be solved directly.
To handle that, an equivalent max-min problem is reformulated, and then an efficient iterative algorithm is proposed.Specifically, the problem is split into three subproblems.For the first two subproblems, they are convex and can be solved using the general optimization toolbox.For the last subproblem, we first transform its non-convex behavior into the convex one by adopting a successive convex approximation technique.Then, these three subproblems are alternately updated in each iteration.Furthermore, we show that the proposed algorithm guarantees the convergence.Finally, simulation results validate that our proposed scheme could achieve a good secrecy performance.

C. Organization and Notations
The rest of the paper is organized as follows.Section II presents the system model and problem formulation.Section III proposes power allocation for secure aerial CoMP.In Section IV, simulation results and discussions are presented.Finally, conclusions are made in Section VI.
Throughout this paper, upper case and lower case boldface letters represent the matrices and the vectors, respectively.I L is an L × L identity matrix, and 0 is a zero vector.E(•) denotes the expectation operation.(•) H and Tr(•) represent the conjugate transpose and the trace of a matrix, respectively.A 0 denotes that A is a positive semidefinite matrix.y ∼ N (0, a) denotes the Gaussian random variable with mean 0 and variance a. x ∼ CN (s, Σ) is the complex circularly symmetric Gaussian distribution with the mean vector s and the covariance matrix Σ.

A. System Model
We consider the downlink transmission in the UAV swarmenabled aerial networks.As illustrated in Fig. 1, the system consists of L UAVs (indexed with 1, ..., L), N legitimate users (indexed with 1, ..., N ) as Bob, and one eavesdropper as Eve.All the legitimate users and the eavesdropper are equipped with N B and N E antennas, respectively.For the UAVs, they form a UAV swarm via CoMP, and act as the aerial base station to assist the wireless networks.
Due to the limited weight and size, only one single antenna is equipped at each UAV.The swarm flies above the coverage area of the legitimate users.In this paper, we assume the same consecutive period, denoting as T U , for each UAV in the swarm, which mainly consists of two parts: flying duration and transmission duration.Here, we assume only during the transmission duration could the UAV swarm hover to serve the legitimate receivers.Furthermore, each legitimate receiver is assumed to be served at most once during the consecutive period.Denote the transmission durations for N legitimate users as τ 1 , τ 2 ,...,τ N , respectively.For simplicity, suppose that the transmission durations are the same, i.e., In addition, suppose that the coordinate of the lth UAV in the nth transmission duration is 2 , where ) and h l [n] denote the horizontal coordinate and the altitude of the lth UAV, respectively.
Over the flight of the UAV swarm, all the legitimate users could be provided with the confidential messages once they are scheduled 3 .Due to the openness of the wireless link, there exists a leakage of the confidential messages.In this system, we assume the eavesdropper is passive and only intends for the confidential messages which are transmitted to the scheduled legitimate users.Furthermore, the eavesdropper randomly moves following the specific trajectory of the swarm to improve eavesdropping.Meanwhile, the eavesdropper also tries to keep a safe distance from the scheduled legitimate users so that it could not be spotted.
We denote the coordinate of the scheduled legitimate user/eavesdropper in the nth transmission duration as (r q [n], t q [n], 0), where q ∈ {B, E}.The locations of the legitimate users/eavesdropper are assumed to be known by the UAV swarm for transmission resource allocation.Thus, the distance between the lth UAV and q at the nth transmission duration is To be practical, we consider both line-of-sight (LoS) and non-line-of-sight (NLoS) connections between the UAVs and the legitimate users.Therefore, the large-scale path loss between the lth UAV and q at the nth transmission duration can be modeled as [36] PL dB q,l [n] = where η LoS , η NLoS , a and b are constants related to the propagation environment, f is the carrier frequency, and c is the speed of light [36].Consequently, the absolute power loss between the lth UAV and q at the nth transmission duration can be expressed as: The channel from the lth UAV to q at the nth transmission duration can be rewritten as where s q,l [n] ∈ C Nq×1 represents the small-scale fading between the lth UAV and q, of which the entries are independently and identically distributed (i.i.d) according to CN (0, 1).
In order to degrade the eavesdropper's channel, each UAV transmits the confidential message in conjunction with AN.Denoting x l [n] as the transmission signal from the lth UAV to the scheduled legitimate user at the nth transmission duration, we have where x s l [n] and x a l [n] represent the confidential message and AN from the lth UAV, respectively.
Furthermore, we express the transmission power from the lth UAV to the scheduled legitimate user at the nth transmission duration as where denote the power of the confidential message and that of the artificial noise transmitted by the lth UAV for the scheduled legitimate user at the nth transmission duration, respectively.
Since each UAV has the limited power, we have where (7) represents the transmission power constraint and P max is the transmission power budget of each UAV.
Considering the energy limitation of the UAVs within the flying period, the following constraint is achieved where (8) denotes the total energy constraint at each UAV over the whole flight, and E max is the energy budget per UAV.
Based on the aforementioned analysis, all the UAVs work together to transmit the confidential messages for the legitimate users, which could form a virtual N q × L MIMO communication link.Note that to avoid the collision, we assume the UAVs are restricted to fly following their specific trajectory with a minimum safety distance between them.In this case, the composite channel matrix H q [n] ∈ C Nq×L between the swarm and q at the nth transmission duration can be expressed as where The received signal at the scheduled legitimate user, denoting y B [n], in the nth transmission duration, and that at the corresponding eavesdropper, denoting y E [n], in the nth transmission duration are given by

B. Problem Formulation
In this subsection, we focus on the problem formulation for this system.Based on (10), the achievable ergodic rate for the scheduled legitimate user at the nth transmission duration is given by and Based on (11), the achievable ergodic rate for the eavesdropper who is intended for the confidential message of the scheduled legitimate user at the nth transmission duration is where Then, the ergodic secrecy rate for the UAV swarm-enabled aerial networks is defined as [ where [x] + = max(0, x). 4 Different from the existing literatures [37], [38], AN in this work is designed by using only the large-scale CSI instead of the instantaneous legitimate CSI.
Owing to the broadcast nature of the wireless channel, AN unavoidably has a leakage and harms the legitimate receivers.Thus, it's important to carefully design the power allocation between the confidential messages and AN so as to minimize the harmful effect on the legitimate users while jamming the eavesdropper, which would be presented in details in the following. 5Here, we assume the noise variance is the same, i.e., equal to δ 2 , over the flying period.For convenience, we drop n here.
In this work, our goal is to maximize the ergodic secrecy rate over the flying period of the UAV swarm by jointly optimizing the power of the confidential messages (i.e., Φ s ) and AN power (i.e., Φ a ) under the constraint of the energy budget for each UAV.The optimization problem can be formulated as It can be observed that problem ( 17) is challenging to be solved for two reasons.First, the operator [•] + results in a nonsmooth manner.Second, even without [•] + , the objective function (17a) has integrals with the expectation operator E(•), which is intractable and difficult to achieve an explicit expression in terms of Φ s and Φ a .

III. POWER ALLOCATION FOR SECURE AERIAL COMP
In this section, we devote our effort to achieve the optimal solutions of problem (17).Before the further analysis, we first handle the nonsmooth of the objective function in problem (17) by adopting the similar analysis in [19], which can be reformulated into max Φs,Φa where problem (18) and problem (17) share the same optimal solution.
To achieve the efficient power allocation, an explicit expression of the objective function in (18) is necessary.Although some works have provided an insightful result to obtain the analytical expression for the objective function, it is generally too cumbersome to do the further power allocation design since the analytical result involves a series of integrals [40].
In the following, we first achieve the closed form of the ergodic secrecy rate in terms of Φ s and Φ a by removing the expectation operator E(•) based on [41].Then, we reformulate the optimization problem.Finally, a computationally efficient iterative algorithm is proposed for the problem and its convergence is presented.
A. Problem Transformation ∀n, the ergodic secrecy rate R Φ s , Φ a can be equivalently rewritten as It can be observed that ( 19) is still intractable due to the expectation operator E(•).To cope with that, we try to approximate (19) by introducing the following theorem.

Theorem 1. By introducing auxiliary variables t
where G(Φ s , Φ a , t B,u , t B,a , t E,u , t E,a ) is defined in (21).Furthermore, G(Φ s , Φ a , t B,u , t B,a , t E,u , t E,a ) is convex in terms of (t B,u , t E,a ), and concave with respect to (t B,a , t E,u ).
Proof.Please refer to Appendix A.
Note that the performance gap between R Φ s , Φ a and R as Φ s , Φ a can be negligible according to [41], which denotes R as Φ s , Φ a is a quite accurate approximation for R Φ s , Φ a .In this case, the optimization problem (18) can be reformulated as

B. An Iterative Algorithm to Solve the Problem
In this subsection, we propose an efficient iterative algorithm to achieve the optimal solutions of problem (22).We first decouple the optimization variables into the following three blocks (t B,u , t E,a ), (t B,a , t E,u ), (Φ s , Φ a ), and then alternately optimize these three blocks one by one by taking the other variables as the constants obtained in the last iteration.Specifically, for any given power indicators (Φ s , Φ a ), the auxiliary variables (t B,u , t E,a ) (or (t B,a , t E,u )) can be efficiently solved through standard algorithm [42].For any obtained auxiliary variables (t B,a , t E,u ) and (t B,u , t E,a ), the power indicators (Φ s , Φ a ) can be optimized by the successive convex approximation technique.
Denote m ≥ 1 as the number of the iteration step.Problem ( 22) can be separated into the following three subproblems.
(1) The optimal variables (t B,u , t E,a ) In the mth iteration, we first optimize (t B,u , t E,a ) with ) obtained in the m − 1th iteration.In this case, problem (22) can be reformulated into Based on Theorem 1, we know that problem ( 23) is convex, which can be efficiently solved by means of the standard optimization toolbox, i.e., CVX.
(2) The optimal variables (t B,a , t E,u ) Based on the obtained ) in the m − 1th iteration, the variables (t B,a , t E,u ) can be achieved by min tB,a,tE,u (24a) which is convex and can be directly solved using CVX.
(3) The optimal variables (Φ s , Φ a ) In the last step of the mth iteration, the variables (Φ s , Φ a ) with the obtained (t m B,u , t m E,a ) and (t m B,a , t m E,u ) can be achieved by solving the following problem max Φs,Φa [n] are concave with respect to P a [n] [41].Thus, the objective function in (25) actually mixes the addition and subtraction of these four concave terms, which is neither concave with respect to Φ s nor concave with respect to Φ a .That is, problem (25) is not convex in terms of Φ s and Φ a .
To overcome the convexity issue, we approximate the second and the third terms of the objective function in (25) to an affine function based on the first-order Taylor expansion.
The gradient of g P a where ϕ B,a [n] = NB δ 2 e t B,a [n] .Thus, the first-order Taylor expansion of g P a is a linear function with respect to P a [n]. 6For convenience, we drop m.
(30b) Problem ( 30) is convex in terms of Φ s and Φ a .Therefore, the optimal variables (Φ s , Φ a ) can be solved by utilizing the standard optimization toolbox CVX.
Based on the above analysis, an overall iterative algorithm for problem (17) can be achieved.Specifically, in each iteration, the original problem (17) can be optimized by alternately solving problem (23), problem (24) and problem (25) in an iterative manner.The details of the proposed algorithm can be summarized in Algorithm 1.

C. Convergence Performance Analysis
Based on the analysis in Section III-B, we could achieve an approximation of G(Φ s , Φ a , t B,u , t B,a , t E,u , t E,a ) as shown in (31).Recall that any concave function is upper bounded by its first-order Taylor expansion at a given local point [43].The following upper-bounded expressions hold and where the equalities in ( 32) and ( 33) are met when Ḡ Φs, Φa, Algorithm 1 The proposed iterative algorithm for solving problem (17) 1: Initialize: the power of the confidential messages and the accuracy ǫ > 0. Set m = 0. Obtain the optimal set (Φ m s ,Φ m a ) by solving problem (25) with the obtained set (t m B,u , t m E,a ) and (t m B,a , t m E,u ).

6:
m − 1 ← m. 7: until The fractional increase of the objective function is below the threshold ǫ > 0.
Thus, we could achieve where the equality holds when Φ s = Φs and Φ a = Φa .Based on the fact (34), we present the convergence of Algorithm 1, as shown next.In the mth iteration, the optimal solutions (Φ m s , Φ m a ), (t m B,u , t m E,a ) and (t m B,a , t m E,u ) can be obtained by Algorithm 1.Based on the properties of the saddle point [42], the following relationship in the mth iteration holds . Then, it follows from (35) that According to (20), we know that7 where step (a) holds since (t m B,a , t m E,u ) are the optimal solutions by using Algorithm 1, and step (b) holds due to the first-order Taylor expansion as shown in (34).
Furthermore, we have Thus, according to (20), (36), (37) and (38), it follows that , tB,u, tB,a, tE,u, tE,a) where step (c) holds since (t B,u , t E,a ) and (t B,a , t E,u ) are the optimal solutions in Algorithm 1, and step (d) holds according to the closed form of the ergodic secrecy rate in (20).(39) indicates that R as Φ s , Φ a is nondecreasing in each iteration, which can assure the convergence of Algorithm 1.

IV. NUMERICAL RESULTS AND DISCUSSIONS
In this section, the performance of our proposed scheme is verified by simulation.We consider a 1000m × 1000m square cell, and suppose there are N = 3 legitimate receivers which are randomly distributed in the cell 8 .The multiple single-antenna UAVs are randomly dispatched in the circular of radius 50m with altitude 100m ∼ 200m, and fly above the coverage area of the scheduled legitimate users.The eavesdropper locates with a safety distance 100m away from the scheduled legitimate user to hide himself, and moves following the trajectory of the UAV swarm for better eavesdropping.Unless otherwise specified, the system parameters are set as follows: the number of antennas for the legitimate users N B = 4, the number of antennas for the eavesdropper N E = 2, the transmission duration τ = 4s, f = 2.4GHz [4], c = 3 × 10 8 m/s, a = 5.0188, b = 0.3511 [36] and the noise covariance δ 2 = −107dBm [41].The threshold presented in Algorithm 1 is fixed as ǫ = 10 −3 .We consider the typical propagation environments using the following (η LoS , η NLoS ) pairs (0.1, 21), (1.0, 20), (1.6, 23), (2.3, 34) corresponding to suburban, urban, dense urban, and highrise urban, respectively [36].
To depict the performance of the proposed scheme, we compare it with the existing scheme in Fig. 2. We assume P 0 s [n] = ps I L , ∀n, where ps = 24dBm, P 0 a [n] = pa I L , ∀n, where pa = 4dBm, E max = 500J and do the simulation for 10 randomly-generated realizations in the suburban environment.In the existing scheme, confidential messages are transmitted for the legitimate receivers by UAVs, and AN is transmitted in the null space of the legitimate channel according to the instantaneous CSI H B via precoding.Similar to [38], we consider the power allocation between the desired signals and AN, where the ratio φ = NB NE +NB of the power budget is allocated to the desired signals, and the ratio 1 − φ is allocated to AN.Note that, this scheme has been widely investigated in the existing literatures but the large-scale CSI has not been taken into account.From the simulation results, we can observe that the proposed scheme presents a significant performance gain 8 This work can be generalized into more legitimate receivers (N > 3).over the existing scheme in the case L = 4 and 6, which can be explained as follows.In our proposed scheme, the UAV swarm could adaptively transmit the confidential messages in a higher power when it is close to the legitimate users based on the large-scale CSI, and allocate the higher power for AN when the swarm is close to the eavesdropper, which promotes the secrecy performance improvement.However, due to the inflexible signal transmission mode in the existing scheme, the confidential messages and AN are transmitted in orthogonal channel spaces.Although the eavesdropper can be well suppressed by exploiting AN, it is not able to improve receiving quality of the legitimate receivers with the fixed power of the desired signals.Therefore, a poor secrecy performance is achieved.
To further illustrate the convergence of Algorithm 1, we present the convergence process for 100 randomly-generated system topologies by the proposed Algorithm 1 in the suburban environment in Fig. 3.We initialize P 0 s [n] = ps I L , ∀n, where ps = 20dBm, P 0 a [n] = pa I L , ∀n, where pa = 0dBm, the power constraint P max = 35dBm, and the energy constraint E max = 500J.It can be observed that for the most cases, Algorithm 1 can converge within 5 iterations, which demonstrates the validity of the proposed scheme.
Fig. 4 illustrates ESR by Algorithm 1 versus different power budget for each UAV in the suburban scenario.We assume P 0 s [n] = ps I L , ∀n, where ps = 24dBm, P 0 a [n] = pa I L , ∀n, where pa = 4dBm, E max = 500J, and achieve ESR based on 10 randomly-generated system topologies.From Fig. 4, we can see that ESR increases when the power budget for each UAV becomes large.That is due to the fact that the increasing transmission power can enhance the achievable ergodic rate at the legitimate user or the eavesdropper.Furthermore, in our proposed scheme, the confidential message and AN are transmitted independently without cooperation at each UAV.Based on the large-scale CSI, each UAV could allocate more power to the confidential message to improve the achievable rate of Bob when the UAV swarm is close to the legitimate users.When the swarm is close to Eve, more power would be allocated to AN for decreasing the signal receiving quality of Eve.Also, for the same power budget of each UAV, ESR increases as the number of UAVs in the swarm.Obviously, the larger the number of UAVs is, the higher the power of the confidential messages is, which significantly enhances the  secrecy performance of the system.Fig. 5 depicts ESR achieved by Algorithm 1 versus total energy budget for each UAV in the suburban scenario, which is derived based on 10 randomly-selected system topologies.It is assumed that P 0 s [n] = ps I L , ∀n, where ps = 10dBm, P 0 a [n] = pa I L , ∀n, where pa = 17dBm, and P max = 35dBm.From Fig. 5, it can be seen that ESR increases as the total energy constraint for each UAV becomes large.That is because when the energy budget at each UAV increases, the transmission power at each UAV is getting large over its flight.By using the large-scale CSI, the power of the confidential messages and the AN power can be intelligently designed.More power can be transmitted for the confidential messages when the swarm is close to Bob, and more power is allocated for AN when the swarm is near to Eve.Thus, a positive ESR can be achieved.Furthermore, we can also see that at the same energy budget of each UAV, ESR grows with the increasing number of UAVs in the swarm.That is due to the fact that as the number of UAVs increases, the total energy of the UAV swarm is getting high.In this case, ESR could increase.
Fig. 6 presents ESR obtained by Algorithm 1 versus the total energy budget for each UAV in different urban scenarios.Here, we achieve ESR based on 10 randomly-generated system topologies.We specify the initial transmission power P 0 s [n] = ps I L , ∀n, where ps = 25dBm, P 0 a [n] = pa I L , ∀n, where pa = 5dBm.It can be observed that ESR is various in the different propagation environments.That comes from that the power loss is highly dependent on the practical urban environments [36], which leads to the divergence of ESR.Also, it can be seen that as the total energy budget for each UAV becomes large, ESR increases.Furthermore, ESR in the case L = 6 is larger than that in the case L = 4.That is because that the total energy of the UAV swarm increases with the number of UAVs, which leads to the higher ESR.

V. CONCLUSIONS
In this paper, we investigated power allocation of AN-assist secure transmission for the UAV swarm-enabled aerial CoMP, where the eavesdropper moved following the trajectory of the swarm for better eavesdropping.We took the composite channel including small-scale and large-scale fading into account.Considering the hardness in acquiring the perfect CSI, we maximized ESR by utilizing only the large-scale CSI of the legitimate receivers and the eavesdropper.Specifically, we designed the problem by jointly optimizing the transmission power of the desired signals and the AN power under the energy constraint of each UAV during its flying period.The formulated problem was a non-convex one.To handle that, we first achieved a closed form of ESR, and then provided an iterative algorithm for the problem.Finally, we evaluated the effectiveness of our proposed iterative algorithm by means of simulation results.

APPENDIX A PROOF OF THEOREM 1
Considering the identical structure of four terms in (19), we focus on the first term and denote it as f B,u (Φ s , Φ a ) = y Φs, Φa, wB,u , (40)

Fig. 1 .
Fig.1.Illustration of a UAV swarm-enabled aerial network, where a UAV swarm, acting as an aerial CoMP, enables MIMO secure communications with the multiple-antenna legitimate users and the eavesdropper in a whole-trajectory-oriented manner.Specifically, the swarm hovers to serve the scheduled legitimate users only in the transmission duration, and the eavesdropper wiretaps the confidential messages by moving following the trajectory of the UAV swarm.

2 : repeat 3 : 1 s, 1 s,
Obtain the optimal set (t m B,u , t m E,a ) by solving problem(23) with the obtained set (Φ m−set (t m B,a , t m E,u ) by solving problem(24) with the obtained set (Φ m−

Fig. 6 .
Fig.6.Ergodic secrecy rate versus total energy budget for each UAV in different typical environments.
respectively, where x s [n] ∼ CN (0, P s [n]) and x a [n] ∼ CN (0, P a [n]) denote the confidential messages and AN transmitted by the UAV swarm at the nth transmission duration4 a ,t E,u , s , Φm−1 a ,